Re: A-invariant

Thanks, Mr. Robert. Especially for clarifying Matrices and Spaces.

Suppose I draw a 3 dimension plane. (Now we switch to plane). The
corner of intersection of orthogonal basis x, y, z is 0.

Now, it is not so clear to me why the intersection of range space and
null space has any dimension from 0 to min(r, n-r).

According to books that I read, an a matrix, M, can only map a vector
to the range space or to the null space.

A vector can be decomposed into the null components and non-null
components (the name is given by me only). The null component is
mapped into 0, that I supposed to be the corner of x,y,z where they
intersect.

The non-null vector is mapped into the range space of M.

Am I correct? or partly correct? Where are the n-r dimension vectors
come from?

Sorry if I misinterprete what you wrote.

Thank you.

Best Regards,

- Boen S. Liong.


So it is only 0 that the intersection of range space and null space.
On 5 Nov, 12:40, Robert Israel <isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
"Boen S. Liong" <mr_bean_cu...@xxxxxxxxx> writes:

A-invariant

Suppose Sn is a space of dimension n. Let Sr = the range space of Sn.
and Snull = the null space of Sn.

Huh? Matrices (or linear transformations) have range spaces and null
spaces. Spaces don't.

What is (vectors and scalars) the intersection of Sr and Snull? Is it
0? Is there any vectors or scalars that can exist both in Sr and
Snull? Is it 0?

All members of vector spaces are vectors, not scalars.
If you're talking about an n x n matrix of rank r, the range has dimension r
and the null space has dimension n-r. The intersection of these could have
any dimension from 0 to min(r, n-r).

Is 0 a scalar or a vector?

There is a scalar 0, and every vector space has a vector 0.

Let Sx be a subspace spanned by vectors x1 through xp that are
orthogonal to each other. p is less than n.
Let Sy be a subspace spanned by vectors y1 through yq that are
orthogonal to each other. q is n-p.
Also yj's are orhogonal to xi's. i = 1,...,p. j= 1,..q.

Is it correct to say Sx is orthogonal to Sy?

Yes. You could also say Sx is the orthogonal complement of Sy.

What is Sx union Sy?

Nothing special.

Is
it Sn? Is there any vectors or scalars that is outside Sx union Sy?

No, and yes. The sum of a nonzero member of Sx and a nonzero member of
Sy is outside the union.

What are the vectors or scalars belong to the intersection of Sx and
Sy? Is it 0?

Yes, the intersection of Sx and Sy consists only of the vector 0.

What are the vectors that is outside the intersection of

Sx and Sy?

All except the vector 0.

Is 0 a vector or a scalar?

Why are you asking this again?
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


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